"His campaign has stabilized after a terrible spring slump," Valliere wrote in a research note.

Bush has been leading most Republican polls of the race, but has faced an insurgency in recent weeks from real-estate magnate Donald Trump, who doubled Bush's support nationally in a poll released Monday by ABC and The Washington Post.

But Valliere, like most analysts, eventually expects Trump to flame out: He gives him a 0% chance of winning the GOP nomination. However, he thinks there's a 35% chance he'll run as a third-party candidate, which would make him the Republican Party's "worst nightmare."

"We've been (sort of) amused by this buffoon, but there was nothing funny about Trump's ugly demagoguery this weekend. He's toast," Valliere wrote, referring to Trump's remarks questioning Sen. John McCain's (R-Arizona) status as a "war hero."

Here are the full odds for Republicans, per Valliere:

Bush: 50%, up from 35%

Wisconsin Gov. Scott Walker: 25%, the same as last month

US Sen. Marco Rubio (R-Florida): 20%, down from 30% last month

Someone else: 5%

And for Democrats, where former Secretary of State Hillary Clinton is still the overwhelming favorite:

Clinton: 80%, the same as last month

Vice President Joe Biden, who has not yet said whether he'll run: 15%. (Valliere thinks there's a 30% chance he will run.)